If a is a (left) topological divisor of zero, then b⁢a is a (left) topological divisor of zero. As a result, a is never a unit, for if b is its inverse, then 1=b⁢a would be a topological divisor of zero, which is impossible.

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In a commutative Banach algebra A, an element is a topological divisor of zero if it lies on the boundary of U⁢(A), the group of units of A.